scholarly journals Stress Concentration and Distribution at Triple Junction Pores of Three-Fold Symmetry in Ceramics

2018 ◽  
Vol 57 (1) ◽  
pp. 63-71 ◽  
Author(s):  
A.B. Vakaeva ◽  
S.A. Krasnitckii ◽  
A.M. Smirnov ◽  
M.A. Grekov ◽  
M.Yu. Gutkin
Keyword(s):  
Stress Concentration ◽  
Triple Junction ◽  
Order Approximation ◽  
Plane Elasticity ◽  
First Order ◽  
First Approximation ◽  
Polycrystalline Ceramic ◽  
Remote Loading ◽  
First Order Approximation ◽  
Polycrystalline Ceramic Material

Abstract The stress concentration and distribution around a triple-junction pore of three-fold symmetry in a polycrystalline ceramic material is considered. The perturbation method in the theory of plane elasticity is used to solve the problem of a nearly circular pore of three-fold symmetry under remote loading in the first approximation. The solution was specified to the uniaxial tension of convex and concave rounded triangular pores. The stress concentration on the pore surface and the stress distribution in vicinity of the pore along its symmetry axes are studied and discussed in detail. The numerical results, issued from the first-order approximation analytical solution, are compared with those of finite-element calculations.

Stokes multipliers for the Orr-Sommerfeld equation

1970 ◽  
Vol 268 (1190) ◽  
pp. 325-349 ◽  
Keyword(s):  
Order Approximation ◽  
Essential Feature ◽  
Neutral Curve ◽  
Stokes Line ◽  
Neutral Stability ◽  
Stokes Phenomenon ◽  
First Order ◽  
First Approximation ◽  
First Order Approximation ◽  
Sommerfeld Equation

The comparison equation method is used to study the outer expansions of the solutions of the Orr-Sommerfeld equation. All but one of these expansions are multiple-valued and must therefore exhibit the Stokes phenomenon. One of the major aims of the present paper is to obtain first approximations to the Stokes multipliers which describe the continuation of these expansions on crossing a Stokes line in the complex plane. By restricting the domains of validity of these expansions appropriately we can insure that all of the expansions are ‘complete ’ in the sense of Olver and this is an essential feature of the work. The resulting approximations show that, in some sectors, a sharp distinction can no longer be made between approximations of inviscid and viscous type. A consistent first-order approximation to the characteristic equation in the complete sense is derived and compared with the more usual second-order approximation of Poincare type. Calculations of the curve of neutral stability for plane Poiseuille flow clearly show that a first approximation in the complete sense provides a substantially better approximation to the neutral curve than a second approximation in the Poincare sense.

A continuum theory of electrostatic probes in a slightly ionized gas

1967 ◽  
Vol 1 (4) ◽  
pp. 407-423 ◽  
Author(s):  
K. Toba ◽  
S. Sayano
Keyword(s):  
Order Approximation ◽  
Debye Length ◽  
Continuum Theory ◽  
Body Dimension ◽  
Ionized Gas ◽  
First Order ◽  
Sheath Region ◽  
First Approximation ◽  
First Order Approximation ◽  
Space Charge Sheath

A systematic continuum theory based on the method of matched asymptotic expansions is developed to deal with electrostatic probes in a slightly ionized gas. To the first approximation of thin space charge sheath, the sheath solution is shown to become common to the geometries considered, i.e. plane, cylindrically and spherically symmetric probe surfaces: the main interest lies in the last geometry. The second-order approximation strongly affects the last two geometries. Detailed numerical calculations are presented to illustrate the effects of a finite ratio of the Debye length to the characteristic body dimension; including a contraction of the sheath region and enhancement of the electric field from the asymptotic (first-order approximation) values.

Improved first-order approximation of eigenvalues and eigenvectors

AIAA Journal ◽  
1998 ◽  
Vol 36 ◽  
pp. 1721-1727
Author(s):  
Prasanth B. Nair ◽  
Andrew J. Keane ◽  
Robin S. Langley
Keyword(s):  
Order Approximation ◽  
Eigenvalues And Eigenvectors ◽  
First Order ◽  
First Order Approximation

Analytical solution for unsteady flow behind ionizing shock wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field

2021 ◽  
Vol 76 (3) ◽  
pp. 265-283
Author(s):  
G. Nath
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Analytical Solution ◽  
Axial Magnetic Field ◽  
Order Approximation ◽  
Ideal Gas ◽  
Field Region ◽  
First Order ◽  
First Order Approximation ◽  
Flow Variables

Abstract The approximate analytical solution for the propagation of gas ionizing cylindrical blast (shock) wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field is investigated. The axial and azimuthal components of fluid velocity are taken into consideration and these flow variables, magnetic field in the ambient medium are assumed to be varying according to the power laws with distance from the axis of symmetry. The shock is supposed to be strong one for the ratio C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ to be a negligible small quantity, where C 0 is the sound velocity in undisturbed fluid and V S is the shock velocity. In the undisturbed medium the density is assumed to be constant to obtain the similarity solution. The flow variables in power series of C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ are expanded to obtain the approximate analytical solutions. The first order and second order approximations to the solutions are discussed with the help of power series expansion. For the first order approximation the analytical solutions are derived. In the flow-field region behind the blast wave the distribution of the flow variables in the case of first order approximation is shown in graphs. It is observed that in the flow field region the quantity J 0 increases with an increase in the value of gas non-idealness parameter or Alfven-Mach number or rotational parameter. Hence, the non-idealness of the gas and the presence of rotation or magnetic field have decaying effect on shock wave.

FIRST-ORDER APPROXIMATION OF THE NUMBER-PROJECTED SO(2N) TAMM-DANCOFF EQUATION AND ITS REDUCTION BY THE SCHUR FUNCTION

1999 ◽  
Vol 08 (05) ◽  
pp. 461-483
Author(s):  
SEIYA NISHIYAMA
Keyword(s):  
Wave Function ◽  
Order Approximation ◽  
Approximate Equation ◽  
Variation Principle ◽  
Schur Function ◽  
Soliton Theory ◽  
First Order ◽  
Fermion Systems ◽  
Group Manifold ◽  
First Order Approximation

First-order approximation of the number-projected (NP) SO(2N) Tamm-Dancoff (TD) equation is developed to describe ground and excited states of superconducting fermion systems. We start from an NP Hartree-Bogoliubov (HB) wave function. The NP SO(2N) TD expansion is generated by quasi-particle pair excitations from the degenerate geminals in the number-projected HB wave function. The Schrödinger equation is cast into the NP SO(2N) TD equation by the variation principle. We approximate it up to first order. This approximate equation is reduced to a simpler form by the Schur function of group characters which has a close connection with the soliton theory on the group manifold.

Chaotic Amplitude Dynamics for Parametrically Excited Systems With Quadratic Nonlinearities

1995 ◽  
Author(s):  
Bappaditya Banerjee ◽  
Anil K. Bajaj
Keyword(s):  
Harmonic Balance ◽  
Chaotic Dynamics ◽  
Degrees Of Freedom ◽  
Order Approximation ◽  
Amplitude Equations ◽  
Analytical Technique ◽  
First Order ◽  
Numerical Studies ◽  
Quadratic Nonlinearities ◽  
First Order Approximation

Abstract Dynamical systems with two degrees-of-freedom, with quadratic nonlinearities and parametric excitations are studied in this analysis. The 1:2 superharmonic internal resonance case is analyzed. The method of harmonic balance is used to obtain a set of four first-order amplitude equations that govern the dynamics of the first-order approximation of the response. An analytical technique, based on Melnikov’s method is used to predict the parameter range for which chaotic dynamics exist in the undamped averaged system. Numerical studies show that chaotic responses are quite common in these quadratic systems and chaotic responses occur even in presence of damping.

scholarly journals Analytical quality assessment of iteratively reweighted least-squares (IRLS) method

2014 ◽  
Vol 20 (1) ◽  
pp. 132-141 ◽  
Author(s):  
Jianfeng Guo
Keyword(s):  
Least Squares ◽  
Order Approximation ◽  
Diagnostic Tools ◽  
Iteratively Reweighted Least Squares ◽  
First Order ◽  
Iterative Reweighting ◽  
First Order Approximation ◽  
Detectable Bias ◽  
Theoretical Analyses ◽  
Exact Analytical Method

The iteratively reweighted least-squares (IRLS) technique has been widely employed in geodetic and geophysical literature. The reliability measures are important diagnostic tools for inferring the strength of the model validation. An exact analytical method is adopted to obtain insights on how much iterative reweighting can affect the quality indicators. Theoretical analyses and numerical results show that, when the downweighting procedure is performed, (1) the precision, all kinds of dilution of precision (DOP) metrics and the minimal detectable bias (MDB) will become larger; (2) the variations of the bias-to-noise ratio (BNR) are involved, and (3) all these results coincide with those obtained by the first-order approximation method.

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